The season's newest political odd couple -- Sen. Edward M. Kennedy and Rep. John B. Anderson -- made back-to-back appearances at the National Urban League today, using similar messages to appeal to black voters but very different styles.

The two presidential hopefuls, who held an extraordinary pow-wow last Thursday, missed each other by only five minutes on the podium but their speeches were drawn from the same well.

Both decried a waning public concern for the problems of black Americans, and both roundly castigated Jimmy Carter, the likely Democratic Party nominee, and Ronald Reagan, the Republican standard bearer.

Anderson, an independent presidential candidate who would like to inherit the support of Kennedy Democrats, drew loud applause when he said the nation needs a "vigorous problem solver, not a tinkerer" like Carter, a "real moral leader, not a moralizer" in the White House.

Kennedy, the long-shot Democratic hopeful, pledged emphatically to continue his battle for an "open" Democratic convention and a more liberal party platform. "What is at stake is not just the name of the Democratic condidate, but the character of the Democratic Party," he said.

Kennedy decried the "right-wing threat" posed by the candidacy of Reagan, scheduled to speak here Tuesday, and the conservative drift of the Democratic Party. Referring to the black woman who touched off more than a decade of protests in the South by sitting across from the front section of a Birmingham bus reserved for whites, he said:

"We will not defeat it [the threat] by tilting toward it. As all of you remember, Rosa Parks did not win her victory for justice by moving to the middle of the bus. And the Democratic Party will not win in 1980 unless we reclaim the issues of economic and social justice." Kennedy said, "there is no substance" behind Reagan's "rhetoric." Anderson said that the platform adopted by the Republican National Convention in Detroit last month offered "black Americans only cutbacks and rhetoric."

The two candidates met only briefly: they passed in the hallway when Kennedy left the convention hall and Anderson entered. But Anderson said he had read Kennedy's remarks and "they are very excellent remarks" -- so much so that they made part of his speech unnecessary.

The flirtation by Anderson, a Republican congressman from Illinois, with Kennedy is part of a calculated effort on his part to court Democratic support. As part of that effort, he has talked in recent days with Kennedy, Rep. Shirley Chisholm (D-N.Y.) former Texas representative Barbara Jordan, officials of New York's Liberal Party, and has scheduled a meeting with Colorado Gov. Richard Lamm in Denver Tuesday.

Anderson, who hinted Friday he would withdraw from the race if Carter lost the nomination, told a press conference today that he did not try to draw Kennedy into an open alliance, but that the two men shared many concerns.

"If Sen. Kennedy is not nominated, I would be willing to sit down and talk about the prospects of him offering some assistance" in the fall campagin, Anderson added.

Buoyed by the results of a national poll by the Louis Harris organization, Anderson said he would rather run against Carter than any other Democrat in the fall because "I think President Carter's the most eminently beatable Democrat."

The Harris survey, released today, reported that Anderson won the support of 25 percent of those polled, compared with 49 percent for Reagan and 23 percent for Carter, in a trial heat that assumed Anderson had a chance to win.

Anderson's style and approach to the Urban League, meeting the week before the Democrats gather here for their national convention, was far different from Kennedy's.

Kennedy has appeared before the Urban League, one of the nation's most prominent civil rights organizations, at three previous national conventions. He and his family have been close to causes advocated by the group for two decades, and today he visited the hospital room of Vernon Jordan, the league's president, who is recuperating from a sniper's gunshot in June.

It is a group Kennedy is comfortable with. And his speech was richly textured with references to his association with past causes and quotations from civil rights leaders.

His attacks on Carter were all placed in a black perspective. At one point, he called the nation's current economic ills "a planned recession." At another, he asked, "What good are federal appointments for black officials if they are told to preside over budget cuts for black people?"

"I'm resolved that the Democratic Party will never become the party that answers, 'No, no, no' to the jobless, homeless, helpless and the hungry," Kennedy said. "We must become again what weVe always been -- the party that says, 'Yes,' to jobs, 'Yes' to housing, 'Yes' to health care and 'Yes' to child nutrition."

As for the GOP nominee, Kennedy declared, "The Ronald Reagan who now talks of jobs for minorities is the same Ronald Reagan who said only two years ago that equal employment opportunity is a 'bureaucratic witch-hunt' . . . He is the same Ronald Reagan who called the 1964 Civil Rights Act a 'bad piece of legislation' and who once joked that 17 million Americans went to bed hungry each night only because they were all on a diet." a

Anderson, whose congressional district is predominantly white, is far less comfortable with black audiences and has had trouble building enthusiasm for his candidacy among blacks. His speech was a programmatic one, a set of proposals for a series of special problems.

For cities, he offered $4 billion annual urban reinvestment trust fund to rebuild decaying roads, bridges and sewer systems. For mass transit problems, he proposed a $4 billion trust fund.

Calling education the greatest problem faced by blacks, he proposed using the new Department of Education to stress a back-to-basics movement, and accountability in the schools.

But Anderson drew one of his loudest applauses with an emotional plea.

"I plead with black Americans, don't mortgage your future to a single party simply because you have been part of a coalition for 50 years," he said. "Why should you be obliged to say, Yes we accept whatever nominee you give us, whether or not we feel in our hearts he is capable of translating our hopes and dreams into something more positive."

Both candidates were received enthusiatically by the large black audience. Kennedy was interrupted by applause 22 times, Anderson, 16.